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A317099
Number of series-reduced planted achiral trees whose leaves span an initial interval of positive integers appearing with multiplicities an integer partition of n.
5
1, 3, 4, 9, 8, 19, 16, 35, 35, 54, 57, 113, 102, 155, 189, 279, 298, 447, 491, 702, 813, 1063, 1256, 1759, 1967, 2542, 3050, 3902, 4566, 5882, 6843, 8676, 10205, 12612, 14908, 18608, 21638, 26510, 31292, 38150, 44584, 54185, 63262, 76308, 89371, 106818, 124755
OFFSET
1,2
COMMENTS
In these trees, achiral means that all branches directly under any given node that is not a leaf or a cover of leaves are equal, and series-reduced means that every node that is not a leaf or a cover of leaves has at least two branches.
EXAMPLE
The a(4) = 9 trees:
(1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
(1112),
(1122), ((12)(12)),
(1123),
(1234).
The a(6) = 19 trees:
(111111), ((111)(111)), (((1)(1)(1))((1)(1)(1))), ((11)(11)(11)), (((1)(1))((1)(1))((1)(1))), ((1)(1)(1)(1)(1)(1)),
(111112),
(111122), ((112)(112)),
(111123),
(111222), ((12)(12)(12)),
(111223),
(111234),
(112233), ((123)(123)),
(112234),
(112345),
(123456).
MATHEMATICA
b[n_]:=1+Sum[b[n/d], {d, Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]], {ptn, IntegerPartitions[n]}];
Array[a, 30]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 01 2018
STATUS
approved